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Zbl 0860.34027
Reinfelds, Andrejs
A reduction theorem for systems of differential equations with impulse effect in a Banach space.
(English)
[J] J. Math. Anal. Appl. 203, No.1, 187-210 (1996). ISSN 0022-247X

The author justifies the reduction principle for the system of weakly nonlinear abstract impulsive equations (1) $x'_i= A_i(t)x_i+ f_i(t,x_1,x_2)$, $\Delta x_i|_{\tau_k}= D_{ik}x_i(\tau_k-0)+ p_{ik}(x_i(\tau_k-0), x_2(\tau_k-0))$, $i=1,2$, $k\in\bbfZ$. Here $(t,x_i)\in \bbfR\times X_i$, $X_i$ ($i=1,2$) are complex Banach spaces, $A_i(t)\in L(X_i)$ $\forall t\in \bbfR$. Particularly, the existence of a unique piecewise continuous bounded integral manifold $M$, which is given by the map $x_2= G(t,x_1): \bbfR\times X_1\to X_2$, is established and the integral distance between an arbitrary solution and $M$ is estimated (Theorem 1). This result allows to prove the global strong dynamic equivalence between (1) and some much simpler (decomposed into two parts) impulsive system (Theorem 2).
[E.Trofimtchouk (Kiev)]
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
34A37 Differential equations with impulses

Keywords: reduction principle; weakly nonlinear abstract impulsive equations; Banach spaces; unique piecewise continuous bounded integral manifold

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